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It is well known that the computation of the Basset-like history force is very demanding in terms of CPU and memory requirements, since it requires the evaluation of a history integral. We use the recent rational theory of Beylkin & Monzón (Appl. Comput. Harmon. Anal., vol. 19, 2005, pp. 17–48) to approximate the history kernel in the form of exponential sums to reformulate the viscous history force in a differential form. This theory allows us to approximate the history kernel in terms of exponential sums to any desired order of accuracy. This removes the need for long-time storage of themore » acceleration histories of the particle and the fluid. The proposed differential form approximation is applied to compute the history force on a spherical particle in a synthetic turbulent flow and a wall-bounded turbulent channel flow. Particles of various diameters are considered, and results obtained using the present technique are in reasonable agreement with those achieved using the full history integral.« less

An explicit expression for the time-dependent force on a stationary, finite-sized spherical particle located in an unsteady inhomogeneous ambient flow is presented. The force expression accounts for both viscous and compressible effects. Towards this end, a time-harmonic plane travelling wave of a given frequency propagating in a viscous compressible flow over a sphere is considered. Linearized compressible Navier–Stokes equations are solved to obtain an analytical expression for the force exerted on the particle in the frequency domain. The force obtained in the Laplace space due to a travelling wave of a given frequency and wavenumber is then generalized to anymore » arbitrary incoming flow. This is achieved by relating the radial and tangential velocity components in the Laplace space to the surface-averaged radial velocity and volume-averaged velocity vectors respectively in the time space. Moreover an expression relating the surface-averaged radial velocity and volume-averaged velocity vector has been provided. The total force is written as a summation of the undisturbed and disturbed force (quasi-steady, inviscid-unsteady and viscous-unsteady) contributions. The force contributions thus obtained are expressed as comprising of two parts – that arising due to spatial variation in the ambient flow and the other arising due to temporal variation. The current formulation is applicable to inhomogeneous ambient flows, however in the limit of negligible Reynolds and Mach numbers. The results are applicable even for particles of sizes larger than the acoustic wavelength. The accuracy of the explicit time-domain force expression is first tested by computing the force on an 80 mm diameter particle due to a weak planar expansion fan. Extension of this formulation when nonlinear effects become important is also proposed and tested by considering strong expansion fans. The results thus obtained are compared against corresponding axisymmetric numerical simulations.« less

In this paper, a force formulation to compute the axial acoustic mean second-order force on finite-sized compressible and rigid particles is presented. The flow inside and outside the spherical inclusion is considered viscous and compressible. Other than for volumetric pulsations of the bubble/droplet, the sphericity of the inclusion is maintained (taken to be unity). A far-field derivation approach has been used to compute the force due to standing and traveling waves; and the force is expressed as a multipole expansion (infinite series). In case of a bubble and a rigid particle, there exist three length scales that govern the meanmore » second-order force: mean radius of the spherical inclusion (R₀), wavelength of the incoming acoustics (λ), and the momentum diffusion thickness of the ambient fluid (δ^o). While R₀ and λ are arbitrary, we assume the viscous length scale is negligibly small compared to the acoustic wavelength. In case of a droplet, however, the following additional parameters (inside to outside fluid ratios) also play a role: density ratio ($$\sim\atop{p}$$), viscosity ratio ($$\sim\atop{μ}$$, and speed of sound ratio ($$\sim\atop{c}$$). The force expression yields the correct behavior in several limiting cases considered: (i) inviscid bubble and droplets with R₀/λ « 1, (ii) inviscid bubbles with finite R₀/λ, and (iii) finite size rigid immovable particles. In general, while the monopole alone is sufficient to capture the force for small bubbles, higher-order terms are found to be important when R₀/λ ≥ 0.02. In addition to reporting similar behavior for droplets, we study the effect arising from $$\sim\atop{p}$$, $$\sim\atop{μ}$$, $$\sim\atop{c}$$, and δ^o/R₀ on the mean second-order force.« less